Issues in the Development of Global Optimization Algorithms for Bilevel Programs with a Nonconvex Inner Program

Transcription

1 Issues in the Development of Global Optimiation Algorithms for Bilevel Programs with a Nonconvex Inner Program Alexander Mitsos and Paul I. Barton* Department of Chemical Engineering Massachusetts Institute of Technolog, , 77 Massachusetts Avenue Cambridge, MA mitsos@mit.edu and pib@mit.edu tel: , fax: Februar 13, 2006 Updated: October 16, 2006 Abstract The co-operative formulation of a nonlinear bilevel program involving nonconvex functions is considered and two equivalent reformulations to simpler programs are presented. It is shown that previous literature proposals for the global solution of such programs are not generall valid for nonconvex inner programs and several consequences of nonconvexit in the inner program are identified. In particular, issues with the computation of lower and upper bounds as well as with the choice of branching variables in a branch-andbound framework are analed. This analsis las the foundation for the development of rigorous algorithms and some algorithmic expectations are established. Ke-words Bilevel program; nonconvex; global optimiation; GSIP; branch-and-bound AMS Classification 65K05=mathematical programming algorithms 90C26=nonconvex programming 90c33=complementarit programming 90C31=sensitivit,stabilit,parametric optimiation 1

2 1 Introduction Bilevel optimiation has applications in a variet of economic and engineering problems and several optimiation formulations can be transformed to bilevel programs. There are man contributions in the literature and the reader is directed to other publications for a review of applications and algorithms [10, 24, 4, 14, 26, 17, 18]. Here, inequalit constrained nonlinear bilevel programs of the form min x, f(x,) s.t. g(x,) 0 arg min h(x,) (1) s.t. p(x,) 0 x X, Y, are considered. The co-operative (optimistic) formulation [14] is assumed, where if for a given x the inner program has multiple optimal solutions, the outer optimier can choose among them. As has been proposed in the past, e.g., [11, 2], dumm variables ( instead of ) are used in the inner program since this clarifies some issues and facilitates discussion. A minimal assumption for an algorithmic development based on branch-and-bound (B&B) is compact host sets X R nx, Y R n and continuous functions f : X Y R, g : X Y R ng, h : X Y R and p : X Y R np. For some of the methods discussed here further assumptions, such as twice continuousl differentiable functions or constraint qualifications, are needed. Note that equalit constraints have onl been omitted for the sake of simplicit and would not alter anthing in the discussion of this paper. 2

3 For a fixed x we denote min h(x,) s.t. p(x,) 0 (2) Y, the inner program. The parametric optimal solution value of this program is denoted w(x) and the set of optimal points H(x). We focus on nonconvex inner programs, i.e., the case when some of the functions p(x,), h(x,) are not partiall convex with respect to for all possible values of x. Most of the literature is devoted to special cases of (1), in particular linear functions, e.g., [9]. Tpicall, the inner program is assumed convex satisfing a constraint qualification and is replaced b the equivalent KKT first-order optimalit conditions; the resulting single level mathematical program with equilibrium constraints (MPEC) violates the Mangasarian-Fromovit constraint qualification due to the complementarit slackness constraints [29]. Fortun-Amat and McCarl [15] reformulate the complementarit slackness conditions using integer variables. Stein and Still [29] consider generalied semi-infinite programs (GSIP) and use a regulariation of the complementarit slackness constraints. We will discuss how this strateg could be adapted to general bilevel programs (1). Ver few proposals have been made for bilevel programs with nonconvex functions. Clark and Westerberg [12] introduced the notion of a local solution where the inner and outer programs are solved to local optimalit. Gümüs and Floudas [17] proposed a B&B procedure to obtain the global solution of bilevel programs; we will show here that this procedure is not valid in general when the inner program (2) is nonconvex. For the case of GSIP an algorithm based on B&B and interval extensions has been recentl proposed b Lemonidis and Barton [20]. We first discuss what is a reasonable expectation for the solution of nonconvex bilevel programs based on state-of-the-art notions in global optimiation. To anale some of the consequences of nonconvexit in the inner program we first introduce two equivalent reformulations of (1) as a simpler bilevel program and 3

4 as a GSIP. We then discuss how to treat the inner variables in a B&B framework and identif issues with literature proposals regarding lower and upper bounds. Finall for KKT based approaches we discuss the need for bounds on the KKT multipliers. 2 Optimalit Requirement An interesting question is how exactl to interpret the requirement arg min Y,p(x,) 0 h(x,) which b definition means that an feasible is a global minimum of the inner program for a fixed value of x. Clark and Westerberg [12] proposed the notion of local solutions to the bilevel program (1) requiring that a furnished solution pair ( x, ȳ) satisfies local optimalit in the inner program. This is in general a ver strong relaxation. Note though that in single-level optimiation finitel terminating algorithms guaranteeing global optimalit exist onl for special cases, e.g., linear programs. For nonconvex nonlinear programs (NLP) f = min x f(x) s.t. g(x) 0 (3) x X state-of-the-art algorithms in general onl terminate finitel with an ε f -optimal solution. That is, given an ε f > 0 a feasible point x is furnished and a guarantee that its objective value is not more than ε f larger than the optimal objective value f( x) f + ε f. Moreover, to our best knowledge, no algorithm can provide guarantees for the distance of the points furnished from optimal solutions ( x x, where x is some optimal solution), which would allow a direct estimate of the requirement arg min. Finall global and local solvers for (3) tpicall allow an ε g violation of the constraints [25]. 4

5 We therefore propose that for bilevel programs (1) with nonconvex inner programs, it is onl plausible to expect a finitel terminating algorithm to provide guarantees that the furnished pair ( x, ȳ) satisfies the constraints of the inner and outer program, ε h -optimalit in the inner program, and ε f -optimalit in the outer program g( x,ȳ) 0 p( x,ȳ) 0 (4) h( x,ȳ) w( x) + ε h f( x,ȳ) f + ε f, where w( x) is the optimal solution value of the inner program for the given x and f is the optimal solution value of the bilevel program (1). Note that as a consequence of ε h -optimalit in the inner program, f f( x,ȳ) need not hold (as Example 2.1 shows). Example 2.2 illustrates that the same behavior can be observed in nonconvex (single level) NLPs when constraint violation is allowed. Example 2.1 (Consequences of ε-optimalit in the Inner Program). Consider the bilevel program min s.t. arg min 2 (5), [ 1 + δ, 1], for some small δ > 0. The inner program gives = 1 and therefore the unique optimal solution is = 1 with an optimal objective value of 1. Assume that the optimalit tolerance of the inner program ε h is such that ε h 2 δ δ 2. Based on definition (4) all [ 1 + δ, 1 ε h ] and [ 1 ε h, 1] are admissible and as a consequence an objective value of 1 + δ can be obtained for the bilevel program (5). Example 2.2 (Consequences of ε-feasibilit in Nonconvex Nonlinear Programs). Consider the nonconvex 5

6 nonlinear program min x x 1 x 2 0 (6) x [ 1 + δ, 2], for some small δ > 0. There are infinitel man feasible points x [1, 2] and the problem satisfies the linear/concave constraint qualification [6, p. 322]. The unique optimal solution is x = 1 with an optimal objective value of 1. Assume that the feasibilit tolerance of the nonlinear constraint ε g is such that ε g 2 δ δ 2. All x [ 1 + δ, 1 ε g ] and x [ 1 ε g, 2] are admissible and as a consequence an objective value of 1 + δ can be obtained for (6). 3 Reformulations One of the major difficulties associated with the solution of (1) is that the two optimiation programs communicate with each other through a solution set H(x), possibl infinite. To facilitate the following discussion, two equivalent optimiation formulations are introduced that communicate with each other through a solution value. These formulations have been proposed in the past b other authors [13, 14, 3] and are also similar to the difference function discussed b Amouegar [2], where convexit is assumed and the variables do not participate in the outer constraints. B the introduction of an additional variable w and the use of the dumm variables, the bilevel program 6

7 (1) is equivalent [13, 14] to min f(x,) x,,w s.t. g(x,) 0 p(x,) 0 h(x,) w 0 (7) x X Y w R w = min h(x,) s.t. p(x,) 0 Y. What is achieved with the introduction of the extra variable w is that the outer and inner program are coupled b a much simpler requirement, namel the optimal solution value of an optimiation program (min) as opposed to the set of optimal solutions (arg min). In principle (7) could be solved using multi-parametric optimiation as a subproblem. As a first step a global multi-parametric optimiation algorithm would be used for the solution of the inner program for all x X dividing X into regions where the minimum w(x) is a known (smooth) function. The second step would be a global solution of the outer program for each of the optimalit regions. This procedure is not suggested as a solution strateg since it would be overl computationall intensive. Note that application of this solution strateg to the original bilevel program (1) would require a parametric optimiation algorithm that furnishes all optimal solutions as a function of the parameter x, and to our best knowledge this is not possible at present. The simplified bilevel program (7) allows the following observation: a relaxation of the inner program results in a restriction of the overall program and vice-versa. This observation also holds for other programs with a bilevel interpretation such as GSIPs, see [21], but does not hold for the original bilevel program 7

8 (1) because an alteration of the inner program can result in an alteration of its set of optimal solutions and as such to the formulation of an unrelated optimiation problem. We will show though that since relaxing the inner program in (7) results in an underestimation of w, the constraint h(x,) w 0 becomes infeasible. Indeed, consider an x and the corresponding optimal objective value of the inner program (2) w. B the definition of optimalit we have h( x,) w for all feasible in the inner program, i.e., Y, s.t. p( x,) 0. As a consequence no exists that can satisf all the constraints of (7) exactl if w is underestimated. If ε h optimalit of the inner program is acceptable and the magnitude of the underestimation is less than ε h it is possible to obtain (approximatel) feasible points of (7). The bilevel programs (1) and (7) are also equivalent to the following GSIP [3] min x, f(x,) s.t. g(x,) 0 p(x,) 0 (8) x X Y h(x,) h(x,), Ȳ (x) Ȳ (x) = { Y : p(x,) 0}. Note that this reformulation does not contradict the observation b Stein and Still [28] that under certain assumptions GSIP problems can be viewed as a special case of bilevel programs. Stein and Still compare a GSIP with variables x to a bilevel program with variables x and whereas the bilevel considered here (1) and the GSIP (8) have the same set of variables x,. Moreover, the GSIP (8) does not necessaril satisf the assumptions made b Stein and Still [28]. Note also that the GSIP (8) does not contain an Slater points as defined in Lemonidis and Barton [21] unless ε h optimalit of the inner program is allowed. In either of the two reformulated programs our proposal of ε h optimalit can be interpreted as ε h feasibilit of the coupling constraint. As mentioned above, NLP solvers tpicall allow such a violation of ordinar nonlinear constraints, and this motivates again the proposal of ε-optimalit (4). 8

9 4 Branching on the Variables As mentioned in the introduction, the main focus of this publication is on the consequences of nonconvex inner programs for B&B procedures. In this section we briefl review some features of this algorithm class and discuss complications for the inner variables. For more details on B&B procedures the reader is referred to the literature, e.g., [19, 30]. For discussion purposes consider again the inequalit constrained nonconvex nonlinear program (NLP) (3). The goal of B&B is to bracket the optimal solution value f between an upper bound UBD, tpicall the objective value at a feasible point, and a lower bound LBD, tpicall the optimal objective value of a relaxed program. To that end the host set X is partitioned into a finite collection of subsets X i X, such that X = i X i. For each subset i a lower bound LBD i and an upper bound UBD i are established. The partitions are successivel refined and tpicall this is done in an exhaustive matter [19], i.e., the diameter of each subset tends to ero. At each iteration the bounds are updated LBD = min i LBD i and UBD = min i UBD i. Therefore UBD is non-increasing and LBD non-decreasing. The algorithm terminates when the difference UBD LBD is lower than a pre-specified tolerance. For bilevel programs the inner program can have multiple optimal solutions for some values of x and therefore a complication is whether or not to branch on the variables. Gümüs and Floudas [17] do not specif in their proposal if the do, and all their examples are solved at the root node. Amouegar [2] for a special convex case proposes to onl branch on the variables x. The dilemma is that on one hand are optimiation variables in the outer program which suggests that one should branch on them but on the other hand the requirement of optimalit in the inner program requires consideration of the whole set Y without branching. One extreme possibilit is to branch on the variables x X and Y without distinction. Example 4.1 shows that some points ( x,ȳ) X i Y i could be feasible for node i but not for the original program (1) and therefore UBD = min i UBD i cannot be used. The other extreme possibilit is to consider for a given node X i a partition of Y to subsets Y j and keeping the worst subset. This procedure is not correct either as Example 4.2 shows. The introduction of dumm variables for the inner program allows the distinction of between the inner and the outer programs. In a forthcoming publication [23] we propose an algorithm where branching is performed on the variables, 9

10 but not on the variables. An alternative would be to emplo subdivision similar to ideas presented for Semi-Infinite Programs [7, 8]. Example 4.1 (The host set of the inner program cannot be partitioned). Consider the simple linear bilevel program min s.t. 0 (9) argmin, [ 1, 1]. The inner program gives = 1 which is infeasible in the outer program b the constraint 0. If we partition the host set Y = [ 1, 1] b bisection (Y 1 = [ 1, 0] and Y 2 = [0, 1]) we obtain two bilevel programs. The first one min s.t. 0 argmin, [ 1, 0] is also infeasible b the same argument. The second min s.t. 0 argmin, [0, 1] 10

11 gives = 0. This point is not feasible in the original bilevel program (9). Taking the optimal solution of this node as an upper bound would give UBD = 0, which is clearl false. Example 4.2 (The host set Y in the outer program cannot be partitioned keeping the worst subset). Consider the simple linear bilevel program min s.t. argmin 2 (10), [ 1, 1]. The inner program gives = ±1 and therefore the unique optimal solution is = 1 with an optimal objective value of 1. If we partition the host set Y = [ 1, 1] b bisection (Y 1 = [ 1, 0] and Y 2 = [0, 1]) we obtain two bilevel programs. The first one min s.t. argmin 2, [ 1, 0] gives the unique feasible point = 1 with an objective value of 1. The second min s.t. argmin 2, [0, 1] gives the unique feasible point = 1 with an objective value of 1. Keeping the worse of the two bounds would result in an increase of the upper bound. 11

12 5 Upper Bounding Procedure Obtaining upper bounds for bilevel programs with a nonconvex inner program is ver difficult due to the constraint is a global minimum of the inner program and no obvious restriction of this constraint is available. Appling this definition to check the feasibilit of a pair of candidate solutions ( x,ȳ) results in a global optimiation problem for which in general state-of-the-art algorithms onl furnish ε f -optimal solutions at finite termination. Onl recentl cheap upper bounds have been developed for some related problems such as SIPs [7, 8] and GSIPs [21] under nonconvexit. In these proposals the upper bounds are associated onl with a feasible point x and no identification of the inner variables is needed. Cheap upper bounds are conceivable for a general bilevel program but to our best knowledge no valid proposal exists for the case of nonconvex inner programs. On termination, algorithms for bilevel programs (1) have to furnish a feasible pair ( x, ȳ), i.e., values must be given for the inner variables. If the upper bounds obtained in a B&B tree are associated with such feasible points, a global solution to a nonconvex optimiation program has to be identified for each upper bound. This leads to an exponential procedure nested inside an exponential procedure and is being emploed in a forthcoming publication [23]. To avoid the nested exponential procedure a formulation of upper bounds without the identification of feasible points is required. 5.1 Solving a MPEC Locall Does Not Produce Valid Upper Bounds Gümüs and Floudas [17] propose to obtain an upper bound b replacing the inner program b the corresponding KKT conditions, under some constraint qualification, and then solving the resulting program to local optimalit. In this section it is shown that this procedure does not in general generate valid upper bounds. In the case of a nonconvex inner program, the KKT conditions are not sufficient for a global minimum and replacing the inner (nonconvex) program b the (necessar onl) KKT conditions amounts to a relaxation of the bilevel program. Solving a (nonconvex) relaxed program to local optimalit provides no guarantees that the point found is feasible to the original bilevel program; moreover the objective value does not necessaril provide an upper bound on the original bilevel program. This behavior is shown in Example 12

13 5.1. Although this example has a special structure, it shows a general propert. Moreover in the proposal b Gümüs and Floudas [17] the complementarit constraint is replaced b the big-m reformulation and new variables are introduced; some issues associated with this are discussed in Section 7. Example 5.1 (Counterexample for upper bound based on local solution of MPEC). For simplicit a bilevel program with a single variable and box constraints is considered. min s.t. argmin 2 (11), [ 0.5, 1]. B inspection the solution to the bilevel program (11) is = 1 with an optimal objective value 1. The inner program has a concave objective function and linear inequalit constraints and therefore b the Adabie constraint qualification the KKT conditions are necessar [5, p. 187]. The KKT conditions for the inner program are given b λ 1 + λ 2 = 0 λ 1 ( 0.5) = 0 (12) λ 2 ( 1) = 0 λ 1 0 λ 2 0. Sstem (12) has three solutions. The first solution is = 0.5, λ 1 = 1, λ 2 = 0, a suboptimal local minimum of the inner program (h = 0.25). The second solution is = 0, λ 1 = 0, λ 2 = 0, the global maximum of the 13

14 inner program (h = 0). Finall the third solution is = 1, λ 1 = 0, λ 2 = 2, the global minimum of the inner program (h = 1). The one-level optimiation program proposed b Gümüs and Floudas [17] is therefore equivalent to min s.t. { 0.5, 0, 1} (13) and has three feasible points. Since the feasible set is discrete, all points are local minima and a local solver can converge to an of the points corresponding to the solutions of the sstem (12). The first solution = 0.5, with an objective value f = 0.5 and the second solution = 0, with an objective value f = 0 are both infeasible for (11). Moreover the clearl do not provide valid upper bounds on the original program (11). The third solution = 1 with an objective value of f = 1, which has the worst objective value for (13), is the true solution to the original program (11). Note that solving (13) to global optimalit would provide a valid lower bound as described in the next section. 6 Lower Bounding Procedure Lower bounds are tpicall obtained with an underestimation of the objective function and/or a relaxation of the feasible set [19] and a global solution of the resulting program. While global solution is a necessar requirement to obtain a valid lower bound, tpicall the constructed programs are convex and therefore global optimalit can be guaranteed with local solvers. 6.1 Relaxation of the Inner Program Does Not Provide a Valid Lower Bound Gümüs and Floudas [17] propose to obtain a lower bound to the bilevel program (1) b constructing a convex relaxation of the inner program (2) and replacing the convex inner program b its KKT conditions. Consider 14

15 for a fixed x a convex relaxation of (2) and the corresponding set of optimal solutions H c ( x) H c ( x) =arg min h c ( x,) s.t. p c ( x,) 0 (14) Y, where h c ( x, ) and p c ( x, ) are convex underestimating functions of h( x, ) and p( x, ) on Y for fixed x. This procedure results in an underestimation of the optimal solution value of the inner program but tpicall not in a relaxation of the feasible set of the bilevel program (1). The desired set inclusion propert for the two sets of optimal solutions H( x) H c ( x) (15) does not hold in general. It does hold when the inner constraints p( x, ) are convex and h c ( x, ) is the convex envelope of h( x, ). Constructing the convex envelope is in general not practical and Gümüs and Floudas propose to use α-bb relaxations [1]. Example 6.1 shows that convex underestimating functions based on this method do not guarantee the desired set inclusion propert even in the case of linear constraints. In the case of nonconvex constraints one might postulate that the set inclusion propert (15) holds if the feasible set is replaced b its convex hull and the objective function b the convex envelope. This is not true, as shown in Example 6.2. Both examples show that the proposal b Gümüs and Floudas does not provide valid lower bounds for bilevel programs involving a nonconvex inner program. Although the examples have a special structure, the show a general propert. Moreover in the proposal b Gümüs and Floudas [17] the complementarit constraint is replaced b the big-m reformulation and new variables are introduced; issues associated with this are discussed in Section 7. Example 6.1 (Counterexample for lower bound based on relaxation of inner program: box constraints). For 15

16 simplicit a bilevel program with a single variable and box constraints is considered. min s.t. arg min 3 (16), [ 1, 1]. The inner objective function is monotone and the unique minimum of the inner program is = 1. The bilevel program is therefore equivalent to min s.t. = 1 [ 1, 1] with the unique feasible point = 1 and an optimal objective value of 1. Taking the α-bb relaxation for α = 3 (the minimal possible) of the inner objective function, the relaxed inner program is min 3 + 3(1 )( 1 ) s.t. [ 1, 1], which has the unique solution = 0, see also Figure 1. Clearl the desired set inclusion propert (15) is violated. Using the solution to the convex relaxation of the inner program, the bilevel program now is equivalent to min s.t. = 0 [ 1, 1] 16

17 and a false lower bound of 0 is obtained. Note that since the objective function is a monomial, its convex envelope is known [22] 3 / 4 1 / 4 if 1 / 2 h c () = 3 otherwise. B replacing the inner objective with the convex envelope the minimum of the relaxed inner program would be attained at = 1 and the set inclusion propert would hold. 1 0 original function 1 convex envelope 2 3 α-bb underestimator Figure 1: Inner level objective function, its convex envelope and its α-bb underestimator for example (6.1) Example 6.2 (Counterexample for lower bound based on relaxation of inner program: nonconvex constraints). For simplicit a bilevel program with a single variable convex objective functions and a single nonconvex constraint in the inner program is considered. min s.t. argmin 2 (17) s.t , [ 10, 10]. 17

18 The two optimal solutions of the inner program are = ±1, so that the bilevel program becomes min s.t. = ±1 [ 10, 10] with the unique optimal point = 1 and an optimal objective value of 1. Taking the convex hull of the feasible set in the inner program, the relaxed inner program min 2 s.t. [ 10, 10] has the unique solution = 0. Clearl the desired set inclusion propert (15) is violated. Using the solution to the convex relaxation of the inner program, the bilevel program program becomes min s.t. = 0 [ 10, 10]. and a false lower bound of 0 is obtained. Note that using an valid underestimator for the function 1 2 would result in the same behavior as taking the convex hull. 6.2 Lower Bounds Based on Relaxation of Optimalit Constraint A conceptuall simple wa of obtaining lower bounds to the bilevel programs is to relax the constraint is a global minimum of the inner program with the constraint is feasible in the inner program and to solve the resulting program globall. Assuming some constraint qualification, a tighter bound can be 18

19 obtained b the constraint is a KKT point of the inner program. In a forthcoming publication [23] we anale how either relaxation can lead to convergent lower bounds. As mentioned in the introduction, Stein and Still [29] propose to solve GSIPs with a convex inner program b solving regularied MPECs. The note that for nonconvex inner programs this strateg would result to a relaxation of the GSIP. The complementarit slackness condition µ i g i (x,) = 0 is essentiall replaced b µ i g i (x,) = τ 2, where τ is a regulariation parameter. A sequence τ 0 of regularied programs is solved; each of the programs is a relaxation due to the special structure of GSIPs. For a general bilevel program the regulariation has to be slightl altered as can be seen in Example 6.3. The complication is that the KKT points are not feasible points in the regularied program and onl for τ 0 would a solution to the MPEC program provide a lower bound to the original bilevel program. A possibilit to use the regulariation approach for lower bounds would be to use an inequalit constraint τ 2 µ i g i (x,) as in [27]. To obtain the global solution with a branch-and-bound algorithm an additional complication is upper bounds for the KKT multipliers, see Section 7. Example 6.3 (Example for complication in lower bound based on regularied MPEC). Consider a linear bilevel program with a single variable and for simplicit take R as the host set. min s.t. arg min (18), R. s.t. 0 The inner program has the unique solution = 0, and thus there is onl one feasible point in the bilevel program with an objective value of 0. Note that all the functions are affine, and the inner program satisfies 19

20 a constraint qualification. The equivalent MPEC is min s.t. 1 µ = 0 µ 0 0 µ ( ) = 0 R. From the stationarit condition 1 µ = 0 it follows µ = 1 and therefore from the complementarit slackness = 0. As explained b Stein and Still [29] the regularied MPEC is equivalent to the solution of min s.t. 1 µ = 0 µ 0 (19) 0 µ ( ) = τ 2 R, which gives µ = 1, = τ 2 and an objective value of τ 2, which clearl is not a lower bound to 0. Replacing the requirement µ = 0 b τ 2 µ( ) or equivalentl τ 2 /µ in (19) would give a correct lower bound of 0. 20

21 7 Complication in KKT Approaches: Bounds A complication with bounds based on KKT conditions are that extra variables µ (KKT multipliers) are introduced (one for each constraint) which are bounded onl below (b ero). The big-m reformulation of the complementarit slackness condition needs explicit bounds for both the constraints g i and KKT multipliers µ i. Fortun-Amat and McCarl [15] first proposed the big M-reformulation but do not specif how big M should be. Gümüs and Floudas [17] use the upper bound on slack variables as an upper bound for the KKT multipliers which is not correct in general as Example 7.1 shows. Note also that Gümüs and Floudas [17] do not specif how to obtain bounds on the slack variables of the constraints but this can be easil done, e.g., b computing an interval extension of the inequalit constraints. Moreover, for a valid lower bound a further relaxation or a global solution of the relaxations constructed is needed and tpicall all variables need to be bounded [30, 25]. The regulariation approach as in Stein and Still [29] does not need bounds for the regulariation but if the resulting program is to be solved to global optimalit bounds on the KKT multipliers are again needed. For programs with a special structure upper bounds ma be known a priori or it ma be eas to estimate those. Examples are the feasibilit test and flexibilit index problems [16] where all KKT multipliers are bounded above b 1. Alternativel, specialied algorithms need to be emploed that do not require bounds on the KKT multipliers. Example 7.1 (Example with arbitraril large KKT multipliers). Let us now consider a simple example with a badl scaled constraint that leads to arbitraril large multipliers for the optimal KKT point min δ( 2 1) 0 (20) [ 2, 2], where δ > 0. The onl KKT point and optimal solution point is = 1 and the KKT multiplier associated with the constraint δ( 2 1) 0 is µ = 1 2δ. The slack variable associated with the constraint can at most take the value δ. As δ 0, the gradient of the constraint at the optimal solution d(δ(2 1)) d = 2 δ ζ = 2 δ 0 21

22 and µ becomes arbitraril large. For δ = 0 the Slater constraint qualification is violated. 8 Conclusions The bilevel program involving nonconvex functions has been discussed, with emphasis on the solution of programs involving nonconvexit in the inner program within a branch-and-bound framework. To that extent two equivalent reformulations of the bilevel program as a simpler bilevel program and as a GSIP have been presented. As a compromise between local and global solution of the inner program, solution to ε- optimalit in the inner and outer programs has been proposed. Several consequences of nonconvexit in the inner program have been identified, and the validit of literature proposals has been explored theoreticall and through simple illustrative examples. In particular, it was shown that replacing the inner program b its necessar but not sufficient KKT constraints and solving the resulting MPEC locall does not produce valid upper bounds. Similarl, a relaxation of the inner program does not lead to a valid lower bound whereas a relaxation of the optimalit constraint does. In the co-operative formulation the inner variables have to be treated differentl than the outer variables. For KKT based approaches, bounds on the KKT multipliers are needed for a global solution of the formulated subproblems. The issues identified with current proposals available in the literature show the need for the development of a rigorous algorithm for the global solution of bilevel programs involving nonconvex functions in the inner program. Acknowledgments This work was supported b the DoD Multidisciplinar Universit Research Initiative (MURI) program administered b the Arm Research Office under Grant DAAD We would like to thank Binita Bhattacharjee, Panaiotis Lemonidis, Benoît Chachuat, Aja Selot and Cha Kun Lee for fruitful discussions. 22

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